https://file.aiquickdraw.com/mathbot/file/1725407212464773un9zx.jpg

To solve the problem in the image, we need to find the following values for the data set provided:

1. \(\sum x\)
2. \(\sum x^2\)
3. \(\left(\sum x\right)^2\)

### Given Data Set:
\[ -6, -1, -3, 6, 1, 5, -5, 0, 4 \]

### 1. Calculate \(\sum x\):

\[
\sum x = -6 + (-1) + (-3) + 6 + 1 + 5 + (-5) + 0 + 4
\]

Let's calculate it:

\[
\sum x = (-6 - 1 - 3 + 6 + 1 + 5 - 5 + 0 + 4) = 1
\]

### 2. Calculate \(\sum x^2\):

\[
\sum x^2 = (-6)^2 + (-1)^2 + (-3)^2 + 6^2 + 1^2 + 5^2 + (-5)^2 + 0^2 + 4^2
\]

Let's calculate each term:

\[
\sum x^2 = 36 + 1 + 9 + 36 + 1 + 25 + 25 + 0 + 16 = 149
\]

### 3. Calculate \(\left(\sum x\right)^2\):

\[
\left(\sum x\right)^2 = (1)^2 = 1
\]

### Results:
- \(\sum x = 1\)
- \(\sum x^2 = 149\)
- \(\left(\sum x\right)^2 = 1\)

Thus, the correct values are:
- \(\sum x = 1\) 
- \(\sum x^2 = 149\)
- \(\left(\sum x\right)^2 = 1\)

It looks like the error in the image was because the second calculation \(\sum x^2\) should be \(149\), not \(16\).

Do you want more details or have any questions?

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**Related Questions:**

1. How does summation notation work in general for a given data set?
2. What are the differences between \(\sum x\) and \(\left(\sum x\right)^2\)?
3. How would the results change if the data set included more negative values?
4. How is the calculation of \(\sum x^2\) used in variance and standard deviation formulas?
5. Can you explain how \(\sum x\) and \(\sum x^2\) are used in calculating the mean square?

**Tip:** Always double-check your calculations to ensure accuracy, especially when working with summation and squared terms.

For the data set, calculate Σx, Σx^2, and (Σx)^2.

Learn how to calculate Σx, Σx^2, and (Σx)^2 for a given data set. This example covers fundamental algebraic operations involving summation and squared terms. Suitable for high school students (Grades 9-12).

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